Electrostatics

PHYS1000

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Electrostatics: Electric fields and forces Electric charge Atoms are made up of electrons, protons, and neutrons. Protons, in the nucleus, and electrons, in the outer part of the atom, have electric charge: particle proton electron

charge +e −e

SI units 1.602 × 10 −19 C −1.602 × 10 −19 C

The SI unit of electric charge is the coulomb (C). (1 C is a very large amount of charge; we’ll see later why we use such an inconvenient unit of electric charge.) • Usually, there are equal numbers of protons and electrons in an object, and the total charge is zero - the object is electrically neutral. • If there are unequal numbers of protons and electrons, the object will have a non-zero total electric charge. Charge can be moved from one object to another – usually, the electrons move, since they are in the outer part of the atom. One of the objects will gain a positive charge, and the other will gain an equal negative charge. The total charge on the two objects remains the same – electric charge is conserved. Moving electrons from one place to another doesn’t change the total number of electrons and protons. You can easily move electric charge yourself, for example, by rubbing a plastic ruler with cloth.

Electric force We observe a force acting between charged objects: object 1 positive negative positive negative or, • opposite charges attract

• same charge repels

object 2 positive negative negative positive

force repulsive repulsive attractive attractive

Electrostatics

PHYS1000

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Electric force between two point charges If some electric charge is all at a single point in space, we call it a point charge. Electrons and protons are point charges. Coulomb’s Law gives the force acting between two point charges: kq1 q2 r2

F=

where q1 and q2 are the two charges, and r is the distance between them, and k = 9.0 × 10 9 N·m2 /C2 . Note that this is very similar to the law of universal gravitation: FG = Gm1 m2 /r2 . If the force given by Coulomb’s law is: • positive → repulsive force • negative → attractive force We often choose to write Coulomb’s law as: F=

1 q1 q2 4π0 r2

The constant 0 = 8.85 × 10−12 C2 /N·m2 is called the permittivity of free space, or the permittivity constant. Remember: force is a vector, so give direction! Point charges are important, since any charged object is a collection of point charges (the electrons and protons). The electric force between any two objects can be found by adding together the forces between the point charges making them up. Conveniently, we find that the electric force due to a uniformly charged sphere is exactly the same as that of a point charge with the same total charge.

Why can a charged object attract an electrically neutral object? We find that a charged object can attract an electrically neutral object. Try this for yourself: charge an object by rubbing, and try to attract small pieces of paper. Why, since the total charge on the paper is zero, are they attracted? electrons repelled to far end far end is −ve, near end is +ve

− −

− −

+ + + +

shorter distance larger attractive force

longer distance smaller repulsive force

−

−

−

− −

−

−

− −

− −

−

−

Electrostatics

PHYS1000

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Electric field What force would act on a charge if we placed it near other charges? We define the electric field E as F = Eq Note that since the electric force is a vector, the electric field is a vector. Since the electric force due to different charges will From Coulomb’s law, we find that the electric field produced by a point charge is: E=

1 q kq = 2 2 4π0 r r

The charge q here is not the same q that appears in F = Eq! The charge q here produces the electric field, the q in F = Eq is affected by the electric field. The direction of the electric field depends on whether the charge is positive or negative:

fields lines directed away from positive charge

field lines directed towards negative charge

We can also draw electric field diagrams like these for arrangements of two or more charges. Remember that: • electric field lines don’t cross • the closer together the field lines are, the stronger the field • field lines start at positive charges, and finish at negative charges The electric field is more than just a mathematical convenience, it stores energy, and if moving, momentum.

PHYS1000

Electrostatics

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Gauss’ law We can measure the total amount of electric field produced by a charge by the electric flux, the electric field times the area through which it passes. If the field in area are at right angles, the flux Φ E = EA. If the field is parallel to the area, the flux is zero. EXTRA Gauss’ law states that ΦE =

q 0

If we consider a uniformly charged sphere, the field must spread out evenly. At a distance r from the centre of the sphere, the surface area of a Gaussian surface we can draw around the sphere is A = 4π r 2 . Thus, EA = E × 4π r2 = q/0 gives us E=

1 q 4π0 r2

which is the same as the electric field produced by a point charge. We also see that inverse square laws (where force etc is proportional to 1/r 2 ) are a consequence of geometry. Other simple cases where we can find the field using Gauss’ law NOT EXTRA! : • Hollow sphere: outside E=

1 q 4π0 r2

• Hollow sphere: inside E=0 • Line of charge ADVANCED E=

1 λ 2π0 r

where the charge/length is λ C/m • Sheet of charge ADVANCED E= where the charge/area is σ C/m2

σ 20

Electrostatics

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Electric potential ADVANCED Since a force acts between two charges, work is done if they are moved apart or closer together. This means that the electric potential energy changes. For example, in a uniform electric field. w = F∆x = Eq∆x In the same way that it’s convenient to use electric field rather than electric force, it’s also convenient to talk about electric potential, the electric potential energy per coulomb (the electric field is the electric force per coulomb). Again, in a uniform electric field, the change in the potential V is ∆V = E∆x We call ∆V the potential difference between the two points. If we know the potential V, the electric potential energy of a charge q is given by w = q∆V The units of potential are volts (V). (A volt is joule/coulomb.) The electron-volt (eV) is sometimes a convenient (non-SI) unit of energy to use; 1 eV is the electronic charge e times 1 V. The potential due to a point charge is V=

1 q kq = 4π0 r r

Since energy is a scalar (ie not a vector), electric potential is not a vector. This makes it very easy to add the potential produced by a collection of point charges.